As with all myths and legends there is a certain element of accuracy with this association. And it is interesting to explore the differing uses of the broom between olden and modern times.
Fundamentally the broom is seen as a symbol of flight. This shrouds the witch in mystery and intrigue. It also shows how the witch is perceived in society. Rationally we know witches do not use their brooms to soar around the countryside (well I certainly don't unless I'm doing something wrong with my broom?) but the myth of flight has persisted through the ages and the endowment of the witch with the super power of flight is a clear indication that witches both then and now were viewed as possessing great powers.
Flight and the broomstick have got a bit confused through the ages. You can decide for yourself whether you actually believe a witch has ever actually levitated into the skies on her broom. But what is much more clear cut is the fact that many witches believed they could. Witches used a concoction popularly known as a flying potion, normally a mixture of animal fats and potent halluciogenic herbs to produce what we would today identify as an acid trip. (Flying potions are still used today and the recipes are surprisingly similar to those of the middle ages).
Under the effects of these potions witches often experienced the sensation of flying. Other witches would not believe they had the physical capacity to actually fly but would use the broom as the symbol for flight out of their physical bodies as flying was also a term used to refer to astral projection for which many witches used (and still do use) flying potions as an aid.
Brooms represent transport and the ability of the witch to move from place to place. This symbolises the witch as a being of transiense, of moving between places and belonging nowhere. For the modern witch, walking the path between the worlds this is a powerful symbol.
The ability to fly is also the ability to rise above. And this is what a witch strives to do. Her goal is to rise above the day to day humdrum of mundane life and align herself with a greater power which she can channel to forge herself a meaningful existence. The broom represents her ability to transcend not only those stuck in routine but also that part of herself that would otherwise be tethered to mundanity.
Of course, in days long gone, brooms were not bought as a specific tool of magic the way they are today. (And beware of spending serious money on a few twigs. I asked one new age shop what wood the brooms were made of and he didn't know.... Worrying. Easy enough to make your own cheaply with wood that may have a personal resonance for you).
A broom would have been present in every home to sweep the floors. Although in the modern world of hoovers and vacumn cleaners the traditional role of the broom is no longer needed, witches of today still use their brooms to sweep. But now, instead of sweeping actual dirt and debris (we suck that up first in our hoovers!) the broom is used to sweep away negativity, bad vibes and aura remnants. The broom cleans the room spiritually in the same way the hoover does physically.
Brooms have an association with fertility and abundance. This is partly due to their very obvious phallic shape. Insertion of the broom was a popular way to absorb the flying ointments (and was probably very enjoyable for the witch). Of course with Ann Summers providing rampant rabbits aplenty, the modern witch generally doesn't need to use her broom for her sexual desires (I certainly don't, mine is square handled - yow!!) This idea of sexuality and fertility is partly to blame for the misconception of flight. Witches would "ride" their brooms through the crop fields to promote fertility of the land and positive crop growth. This idea of the witches on their brooms morphed into the myths of broom flight. I personally have no modern equivalent (though if my local farmer wants me to ride through a few of his fields, I'd not have a problem with that).
So the broom continues to be a powerful tool for the witch. It symbolises the core of our craft, our ability to transcend, the cleaning of magical clutter in our homes and it provides us with a link to our ancestors, the witches of old.
Image http://www.freedigitalphotos.net/images/Halloween_g164-Halloween_Background_p55837.html (Sujin Jetkasettakorn)